What is (\partial)/(\partial x)(x^{y^x}) ?

The answer to (\partial)/(\partial x)(x^{y^x}) is x^{y^x}(y^xln(x)ln(y)+(y^x)/x)

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(\partial)/(\partial x)(x^{y^x})

\frac{\partial}{\partial x} (x^{y^{x}})

x^{y^{x}} (y^{x} ln (x) ln (y) + \frac{y ^{x}}{x})

Solution steps

\frac{\partial}{\partial x} (x^{y^{x}})

Treat

y

as a constant

Apply exponent rule:

a^{b} = e^{b l n (a)}

x^{y^{x}} = e^{y^{x} l n (x)}

= \frac{\partial}{\partial x} (e^{y^{x} l n (x)})

Apply the chain rule:

e^{y^{x} l n (x)} \frac{\partial}{\partial x} (y^{x} ln (x))

= e^{y^{x} l n (x)} \frac{\partial}{\partial x} (y^{x} ln (x))

\frac{\partial}{\partial x} (y^{x} ln (x)) = y^{x} ln (y) ln (x) + \frac{y ^{x}}{x}

= e^{y^{x} l n (x)} (y^{x} ln (y) ln (x) + \frac{y ^{x}}{x})

e^{y^{x} l n (x)} = x^{y^{x}}

= x^{y^{x}} (y^{x} ln (x) ln (y) + \frac{y ^{x}}{x})

\int \frac{1}{x ^{2} + 64} d x

y = 2 x^{3} - x^{2} + 2

\int \frac{e ^{x^{n}}}{x ^{1 - n}} d x

h \to 0 lim (\frac{5 e ^{x} - 5 e ^{x + h}}{3 h})

\frac{e ^{- s}}{( s ^{2} + 8 )}

What is (\partial)/(\partial x)(x^{y^x}) ?
The answer to (\partial)/(\partial x)(x^{y^x}) is x^{y^x}(y^xln(x)ln(y)+(y^x)/x)